Homological Integrals for Weak Hopf Algebras

TL;DR

This paper extends the theory of homological integrals to infinite-dimensional weak Hopf algebras by defining left and right homological integrals for AS Gorenstein weak Hopf algebras, and showing these integrals are invertible objects that control the Nakayama bimodule via $U \cong \int^{\ell}_H \overline{\otimes}^r H^{S^2}$. It establishes a deep connection between the Van den Bergh condition and Artin–Schelter Gorensteinness in this setting, leading to the bijectivity of the antipode under natural hypotheses. The authors also develop the fundamental relationship between total integrals and dualities, prove that noetherian weak Hopf algebras that are finite over affine centers decompose as finite direct sums of AS Gorenstein weak Hopf algebras, and show each summand satisfies the Van den Bergh condition. These results extend Brown–Goodearl-type structure theorems to the weak Hopf algebra context, providing a robust structural framework and paving the way for further classifications, including GK-dimension one cases.

Abstract

We introduce the notion of a homological integral for an infinite-dimensional weak Hopf algebra and use the homological integral to prove several structure theorems. For example, we prove that the Artin--Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove a decomposition theorem that states that any weak Hopf algebra finite over an affine center is a direct sum of Artin--Schelter Gorenstein, Cohen--Macaulay, GK dimension homogeneous weak Hopf algebras.

Theorems & Definitions (74)

• Definition 1
• Definition 2
• Proposition 3: Proposition \ref{['yypro3.7']}
• Theorem 4
• Theorem 5: Theorem \ref{['yythm4.6']}
• Theorem 6
• Lemma 1.1
• Lemma 1.2
• proof
• Lemma 1.3